This is a common misconception. According to YouTuber Cassidy Cash (That Shakespeare Life), Shakespeare used the word 'vomit' six times in his plays, but he didn't invent it. The word first appeared in Middle English, around the year 1400; dictionary.com is just one source that confirms this.
The misconception may arise from confusion with the word 'puke'. In As You like It (Act II, Scene VII) Jacques, in his "All the world's a stage" speech, refers to "the infant, Mewling and puking in the nurse's arms." Shakespeare is believed to have been the first writer to use this word.
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